Nuprl Lemma : expfact_wf
∀[m:ℕ+]. ∀[k:ℕ]. ∀[n:ℕ+]. ∀b:{b:ℕ| n * k^b < (b)!} . ((m ≤ b) ⇒ (expfact(m;k;n * k^m;(m)!) ∈ {b:ℕ+| (n * k^b) ≤ (b)!} \000C))
Proof
Definitions occuring in Statement :
expfact: expfact(n;x;p;b),
fact: (n)!,
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat_plus: ℕ+,
nat: ℕ,
subtype_rel: A ⊆r B,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
expfact: expfact(n;x;p;b),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
subtract: n - m,
le: A ≤ B,
top: Top,
squash: ↓T,
less_than: a < b,
has-value: (a)↓,
sq_type: SQType(T),
sq_stable: SqStable(P),
true: True
Lemmas referenced :
istype-le,
istype-less_than,
exp_wf2,
fact_wf,
istype-nat,
nat_plus_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
subtract-1-ge-0,
le_int_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
equal-wf-base,
bool_wf,
set_subtype_base,
nat_wf,
less_than_wf,
le_wf,
int_subtype_base,
assert_wf,
lt_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
nat_plus_subtype_nat,
add-zero,
minus-zero,
multiply-is-int-iff,
false_wf,
int_term_value_mul_lemma,
istype-void,
itermMultiply_wf,
decidable__lt,
value-type-has-value,
int-value-type,
subtype_base_sq,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
mul-swap,
subtract-elim,
minus-one-mul,
exp_step,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
zero-add,
fact_unroll_1,
sq_stable__less_than,
true_wf,
squash_wf,
zero-mul,
add-mul-special
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
setIsType,
inhabitedIsType,
multiplyEquality,
applyEquality,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
because_Cache,
dependent_set_memberEquality_alt,
unionElimination,
baseApply,
closedConclusion,
baseClosed,
intEquality,
equalityElimination,
productElimination,
equalityIstype,
promote_hyp,
pointwiseFunctionality,
imageElimination,
callbyvalueReduce,
addEquality,
instantiate,
cumulativity,
minusEquality,
imageMemberEquality
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}].
\mforall{}b:\{b:\mBbbN{}| n * k\^{}b < (b)!\} . ((m \mleq{} b) {}\mRightarrow{} (expfact(m;k;n * k\^{}m;(m)!) \mmember{} \{b:\mBbbN{}\msupplus{}| (n * k\^{}b) \mleq{} (b)!\} ))
Date html generated:
2020_05_19-PM-10_03_36
Last ObjectModification:
2019_12_31-PM-00_11_42
Theory : num_thy_1
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